Theorem zarmxt1 33911

Description: The Zariski topology restricted to maximal ideals is T₁ . (Contributed by Thierry Arnoux, 16-Jun-2024.)

Hypotheses

Ref	Expression
zartop.1	⊢ 𝑆 = (Spec‘𝑅)
zartop.2	⊢ 𝐽 = (TopOpen‘𝑆)
zarmxt1.1	⊢ 𝑀 = (MaxIdeal‘𝑅)
zarmxt1.2	⊢ 𝑇 = (𝐽 ↾t 𝑀)

Assertion

Ref	Expression
zarmxt1	⊢ (𝑅 ∈ CRing → 𝑇 ∈ Fre)

Proof of Theorem zarmxt1

Dummy variables 𝑖 𝑗 𝑘 𝑙 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zartop.1	. . . 4 ⊢ 𝑆 = (Spec‘𝑅)
2		zartop.2	. . . 4 ⊢ 𝐽 = (TopOpen‘𝑆)
3	1, 2	zartop 33907	. . 3 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Top)
4		zarmxt1.1	. . . 4 ⊢ 𝑀 = (MaxIdeal‘𝑅)
5	4	fvexi 6890	. . 3 ⊢ 𝑀 ∈ V
6		zarmxt1.2	. . . 4 ⊢ 𝑇 = (𝐽 ↾t 𝑀)
7		resttop 23098	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ V) → (𝐽 ↾t 𝑀) ∈ Top)
8	6, 7	eqeltrid 2838	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ V) → 𝑇 ∈ Top)
9	3, 5, 8	sylancl 586	. 2 ⊢ (𝑅 ∈ CRing → 𝑇 ∈ Top)
10		eqid 2735	. . . . . . . . . . 11 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
11	10	mxidlprm 33485	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (PrmIdeal‘𝑅))
12	11	ex 412	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (𝑚 ∈ (MaxIdeal‘𝑅) → 𝑚 ∈ (PrmIdeal‘𝑅)))
13	12	ssrdv 3964	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → (MaxIdeal‘𝑅) ⊆ (PrmIdeal‘𝑅))
14	13	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → (MaxIdeal‘𝑅) ⊆ (PrmIdeal‘𝑅))
15		eqid 2735	. . . . . . 7 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
16	14, 4, 15	3sstr4g 4012	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑀 ⊆ (PrmIdeal‘𝑅))
17		sseq2 3985	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑙 → (𝑖 ⊆ 𝑗 ↔ 𝑖 ⊆ 𝑙))
18	17	cbvrabv 3426	. . . . . . . . . . . 12 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑙}
19		sseq1 3984	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝑖 ⊆ 𝑙 ↔ 𝑘 ⊆ 𝑙))
20	19	rabbidv 3423	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑙} = {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑙})
21	18, 20	eqtrid 2782	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑙})
22	21	cbvmptv 5225	. . . . . . . . . 10 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) = (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑙})
23	1, 2, 15, 22	zartopn 33906	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) ∧ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) = (Clsd‘𝐽)))
24	23	adantr 480	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) ∧ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) = (Clsd‘𝐽)))
25	24	simpld 494	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)))
26		toponuni 22852	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) → (PrmIdeal‘𝑅) = ∪ 𝐽)
27	25, 26	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → (PrmIdeal‘𝑅) = ∪ 𝐽)
28	16, 27	sseqtrd 3995	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑀 ⊆ ∪ 𝐽)
29		simpl 482	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑅 ∈ CRing)
30	29	crngringd 20206	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑅 ∈ Ring)
31		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ ∪ 𝑇)
32	6	unieqi 4895	. . . . . . . . . . . 12 ⊢ ∪ 𝑇 = ∪ (𝐽 ↾t 𝑀)
33	31, 32	eleqtrdi 2844	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ ∪ (𝐽 ↾t 𝑀))
34	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝐽 ∈ Top)
35		eqid 2735	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
36	35	restuni 23100	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ⊆ ∪ 𝐽) → 𝑀 = ∪ (𝐽 ↾t 𝑀))
37	34, 28, 36	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑀 = ∪ (𝐽 ↾t 𝑀))
38	33, 37	eleqtrrd 2837	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ 𝑀)
39	38, 4	eleqtrdi 2844	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ (MaxIdeal‘𝑅))
40		eqid 2735	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
41	40	mxidlidl 33478	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (LIdeal‘𝑅))
42	30, 39, 41	syl2anc 584	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ (LIdeal‘𝑅))
43		eqid 2735	. . . . . . . . . 10 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
44	22, 43	zarclssn 33904	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ (LIdeal‘𝑅)) → ({𝑚} = ((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})‘𝑚) ↔ 𝑚 ∈ (MaxIdeal‘𝑅)))
45	44	biimpar 477	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → {𝑚} = ((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})‘𝑚))
46	29, 42, 39, 45	syl21anc 837	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} = ((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})‘𝑚))
47	22	funmpt2 6575	. . . . . . . 8 ⊢ Fun (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
48		fvex 6889	. . . . . . . . . . 11 ⊢ (PrmIdeal‘𝑅) ∈ V
49	48	rabex 5309	. . . . . . . . . 10 ⊢ {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑙} ∈ V
50	49, 22	dmmpti 6682	. . . . . . . . 9 ⊢ dom (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) = (LIdeal‘𝑅)
51	42, 50	eleqtrrdi 2845	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ dom (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
52		fvelrn 7066	. . . . . . . 8 ⊢ ((Fun (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) ∧ 𝑚 ∈ dom (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})) → ((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})‘𝑚) ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
53	47, 51, 52	sylancr 587	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → ((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})‘𝑚) ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
54	46, 53	eqeltrd 2834	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
55	24	simprd 495	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) = (Clsd‘𝐽))
56	54, 55	eleqtrd 2836	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} ∈ (Clsd‘𝐽))
57	38	snssd 4785	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} ⊆ 𝑀)
58	35	restcldi 23111	. . . . 5 ⊢ ((𝑀 ⊆ ∪ 𝐽 ∧ {𝑚} ∈ (Clsd‘𝐽) ∧ {𝑚} ⊆ 𝑀) → {𝑚} ∈ (Clsd‘(𝐽 ↾t 𝑀)))
59	28, 56, 57, 58	syl3anc 1373	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} ∈ (Clsd‘(𝐽 ↾t 𝑀)))
60	6	fveq2i 6879	. . . 4 ⊢ (Clsd‘𝑇) = (Clsd‘(𝐽 ↾t 𝑀))
61	59, 60	eleqtrrdi 2845	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} ∈ (Clsd‘𝑇))
62	61	ralrimiva 3132	. 2 ⊢ (𝑅 ∈ CRing → ∀𝑚 ∈ ∪ 𝑇{𝑚} ∈ (Clsd‘𝑇))
63		eqid 2735	. . 3 ⊢ ∪ 𝑇 = ∪ 𝑇
64	63	ist1 23259	. 2 ⊢ (𝑇 ∈ Fre ↔ (𝑇 ∈ Top ∧ ∀𝑚 ∈ ∪ 𝑇{𝑚} ∈ (Clsd‘𝑇)))
65	9, 62, 64	sylanbrc 583	1 ⊢ (𝑅 ∈ CRing → 𝑇 ∈ Fre)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  {crab 3415  Vcvv 3459   ⊆ wss 3926  {csn 4601  ∪ cuni 4883   ↦ cmpt 5201  dom cdm 5654  ran crn 5655  Fun wfun 6525  ‘cfv 6531  (class class class)co 7405  Basecbs 17228   ↾_t crest 17434  TopOpenctopn 17435  LSSumclsm 19615  mulGrpcmgp 20100  Ringcrg 20193  CRingccrg 20194  LIdealclidl 21167  Topctop 22831  TopOnctopon 22848  Clsdccld 22954  Frect1 23245  PrmIdealcprmidl 33450  MaxIdealcmxidl 33474  Speccrspec 33893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-ac2 10477  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-rpss 7717  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fi 9423  df-dju 9915  df-card 9953  df-ac 10130  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-sca 17287  df-vsca 17288  df-ip 17289  df-tset 17290  df-ple 17291  df-rest 17436  df-topn 17437  df-0g 17455  df-topgen 17457  df-mre 17598  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-submnd 18762  df-grp 18919  df-minusg 18920  df-sbg 18921  df-subg 19106  df-cntz 19300  df-lsm 19617  df-cmn 19763  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-ring 20195  df-cring 20196  df-subrg 20530  df-lmod 20819  df-lss 20889  df-lsp 20929  df-sra 21131  df-rgmod 21132  df-lidl 21169  df-rsp 21170  df-lpidl 21283  df-top 22832  df-topon 22849  df-bases 22884  df-cld 22957  df-t1 23252  df-prmidl 33451  df-mxidl 33475  df-idlsrg 33516  df-rspec 33894

This theorem is referenced by: (None)